Arithmetic and Number Theory Skills
About folder contents
Practical Methods Ends and Values for Arithmetic
The 12 Times Table Visually
The 20 Times Table
Expression Evaluation how to show work
Formula Evaluation - how to show work
Exact Arithmetic - Wholes and Fractions
Quick history of numbers and algebra
§ 1 Decimal Place Value:
§ 2 Arithmetic with Decimals:
§ 3 Prime Factorization Skills:
§ 4 Remainder Arithmetic and Divisibility:
§ 5 Integers:
§ 6 Fractions and Ratios:
§ 7 Arithmetic and Fractions with Units:
§ 8 Arithmetic with Signed Numbers:
§ 9 Combinatorics - Trees Tables and Products:
§ 10 LCM GCD and Euclid GCD Algorithm:
§ 11 Squares and Square Roots:
§ 12 Comparison of Unsigned and Signed Numbers:
arithmetic videos - Real Player Format
Multilevel Skill Mastery
When a young child complains there are too many letters in the alphabet to learn, we may
laugh and say "No, No - You will need to learn all the letters to spell, read and write". Later
when an older child or teenager says there addition or times tables are too complicated,
or that methods for adding, comparing, subtracting, multiplying and dividing decimals and fractions
are too many learn, we should also say "No, No - arithmetic with numbers and with units of measure
has value for daily- and adult-life." We may further explain that numbers, units of measure, units
of counting and maps-plans-diagrams drawn to scale are
everywhere. In particular, all or some are formally or informally involved in time-date-calendar matters; in money matters - saving or buying
and selling; in describing how likely or not is the chance of rain, or the chance of wining or losing; in cooking
or making things; and in navigation - where are we on this map-plan-diagram, or which way do we want to go.
Just as learning to read and write involves mastery of the letters of an alphabet, daily and adult-life involves
counting and arithmetic with numbers alone and with units of counting and measure. In modern life, numbers appear
on street signs, in describing or figuring how many or how much. Check that - look around you.
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Notes for Learners
Most high school mathematics is preparation for calculus-based
college studies in business, science, technology, engineering or
further mathematical subjects. That preparation may begin with
mastery of the following counting, figuring and number skills
and practices. In this preparation, focus on
learning how to do - the explanations where present are intended
to provide links and connections that make operation mastery
easier.
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Decimal place and decimal arithmetic: For place value you
need to the value of each digit before and after a decimal
point. Comparision of numbers given in decimal form requires
place value understanding and so is a test of it. In decimal
arithmetic without a calculator, you need to be able to add
several numbers, subtract two, multiply two and divide two.
Rules for placement of decimal points in result need to be
mastered. In the case of whole numbers, you need master long
division and through it calculate quotients and remainders.
Note: 5 goes into 17, exactly 3 times with a remainder of 2.
Better yet we may note that the remainder 2 is 2 fifths of 5.
So 5 goes into 17 exactly 3 and 2 fifths times. Furthermore,
the latter is equal to the improper fraction 17 fifths as 3
times one is 3 times 5 fifths.
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Prime Factorization Skills: You should be able to recognize
when numbers under 100 are prime or composite. In the case of
composite numbers, you need to be able to find their prime
factorization quickly. Here you are in luck, site coverage of
prime factorization skills provides a simpler explanation of
what is a prime and shows how to use time tables and then use
a square rule to quickly and efficiently obtain the prime
factorization of all whole numbers less than 169 =
132. For that you eventually need to know all
multiples of 11 and 7 under 169, and you need to know how to
recognize using its decimal form, when a whole number is a
multiply of 2, 3 or 5. The square rule is well-known - the
coverage here has taken the liberty of giving it name and of
using it to provide a quick and easy method for recognizing
when whole numbers less than 169 are prime or for obtaining
their prime factorization and
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Fractions: You should know how to add, subtract,
compare and divide fractions with like denominators. Then you
know how to raise terms to add, subtract, compare and divide
fractions with unlike denominators by raising terms to get
common denominators.
Remark: In measuring distance, we may measure
length in terms of centimeters or meters. Here the
denominate number 1.5 meters is 150 centimeters. The latter
two denominate numbers are written differently but have the
same value. Now if a two pies are cut both cut into
quarters, you may take five quarters. If each quarter is
cut is thirds, then 12 one thirds of a quarter has the same
value as one pie - ignore the crumbs. So one third of a
quarter is one twelfth. Now each quarter of a pie equal 3
times a third or itself, that is 3 twelfths. Hence 5
quarters will be the same as 5 times 3 twelfths or 15
twelfths. In answering the question how much pie do you
have, the answer 5 quarters gives the result by counting
quarters while the answer 15 twelfths gives the answer by
counting twelfths. Fractions are used to count how many
same-sized parts of one appear. With fractions, Raising
and lowering terms changes the size of the parts used to
answer the question how much of one or more than one
appears. Raising and lower terms in fractions to obtain a
common denominator is like finding a common unit for
lengths when different lengths are give using different
units: centimeter, meters or kilometers.
Multiplication with words:
One fifth of 20 objects is 4 objects since 20 is the multiple
4 of five. Likewise, 3 fifths of 20 would be 3 times one
fifth of 20 objects, that is 3 × 4 objects or 12 objects.
From the foregoing one fifth of 20 thirds would 4 thirds and
3 fifths of 20 thirds would be 3 times 4 thirds or 12 thirds.
Thus 3 fifths of 20 thirds is easily calculated because the
number of thirds 20 is a multiple of the denominator 5 in the
fraction 3 fiths. In general, products of two fractions, that
is to calculate one fraction of a second fraction, we may
raise terms in the second fraction so that its numerator is a
multiple of the denominator of the first. For example, 8
sevenths of 3 fifths would be 8 sevenths of 21
thirty-fifths, or 8 times 3 thirty-fifths. See site fraction
lessons to redo all the foregoing on paper and more efficiently with arithmetic notation
in place of words.
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You need to know how to add, subtract, compare, multiply and
divide fractions quickly, exactly and efficiently with the
aid of prime number factorization of numerators and
denominators. That is required for skill in further
mathematics from algebra to calculus, even if you prefer to
use a calculator and decimals instead. In the first instance
learning to do is more importance for marks and final
examinations than undertanding how or why arithmetic work.
But the why is a brain stretching exercise strongly
recommended for the sake of calculus mastery later on. In
primary and junior high school mathematics, the take home
value of learning to do is the main objective. But in senior
high school mathematics upto or including, understanding why
methods needs to be valued more and more. You can start here
with study of the why methods work. Perhaps a teacher, a
tutor, a parent or fellow will help.
Site coverage of
Fractions provides a reference for the
introduction of fraction skills and concepts. For reference
if not immediate use in explanation, this coverage show how
raising terms allows addition, comparision, subtraction and
division with unlike denominators can be converted to the
easier and feasible like denominator case. The simple case
for fraction multiplication appears when numerator of a
multiplicand is a multiple of the numerator of a
multiplier. In general, raising terms in the multiplicand
to obtain the simple case provides a path for understanding
and introducing multiplication general. The foregoing
explanations overwhelming for some students, may help skill
mastery for others. For skill perfection of the first kind,
learning how is more important than learning why - the
latter being optional for the immediate take-home value of
skill development. But in the long-term, skill perfection
with understanding of why will make further the
undertanding part of further mathematics easier to grasp -
practice makes perfect.
For efficient fraction skills you should know how to
calculate least common multiples and least common denominators
using [a]a list method for small whole numbers, and [b] the
more power prime factorizatin methods for larger whole numbers
- the list method would require more work, too much work. You
should know how to calculate greatest common divisiors of
numerators and denominators for say, lowering terms in
fractions. Finally, you should be able to answer the question:
What is the likely benefit of using least common
denominators in the addition and subtraction of fractions.
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You need to master arithmetic with integer
and further signed numbers.
Site coverage of integers and further signed numbers appear
three times here: once in arithmetic, once in geometry and
once in algebra. Site coverage gives some rules in slogan
form easy to understand and explain. Site coverage also
includes ideas or lessons on how to provide motivation for
arithmetic rules with signed. [In my school days, the modern
mathematics course design than in use obtained arithmetic
signed number from assumptions about numbers. Here in
contrast, we follow a more concrete and direct route, simpler
to understand and explain - a route that implies the
assumptions.]
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You should master or should be shown Arithmetic with units.
Units appear in measurements and counts of time, distance,
mass, weight and money, etc. The ability to do arithmetic with
numbers and units alone or in fractions has take-home value
- could make some calculation at home and work easier to do.
In senior high school and college science, arithmetic units
may be carried carefully throught calculations. I prefer that
because carrying units through calculations can help in
checking results and can help delay, avoids or eases the
change of units that is sometimes needed. Most likely, you
will see units carried through the calculations by chemistry
and physic while mathematics teachers may follow the
alternate route of doing conversion in a way that eliminates
the appearance of units in calculation. You have too see and
master both approaches. High school mathematics has
traditionally shown student how to calculations with
monomials, calculations that have little or not meaning. The
same or similar calculation skill can be obtained and
practices by mastering the more useful arithmetic with units.
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Notes for Instructors
The Gifted students in your courses may be able to read or digest the
material below. The material was written to provide a reference for skill
development, a reference that emphasizes skill and comprehension.
At this level, giving students methods and patterns to follow to obtain
result in an observable and correctable manner is sufficient for skill
development. Explanations of why or how methods work should be only be
given in class where doing so aids skill development and does not
overwhelm it. A balance is needed. The balance may be different for each
student. Skill perfection of the first kind consists of learning to do
and record steps in a way the doer and others may see as done or later
for checking or correction. This skill perfection implies awareness
of the domino effect of mistakes in multisteps. At work and in school
that awareness is a sign of intelligence of the practical kind. In the
first instance is more important than partial or full comprehension of
why associated with second kind of skill know-how perfection. Skill
perfection of the second kind consists of partial or full understanding
of how and why the steps work.
Since the arithmetic and number theory skills below span skills that
might be met and mastered between the ages of 9 and 14 say, or older if
need-be. No one year of instruction should cover all skills and concepts
below. To set a target for your skill development for students in or
before the first year of secondary school, students 13 years of age, here
some skill development objectives for the latter age group.
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Fill in the 12 times and sum table quickly.
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Be aware of the domino effects of errors in calculation.
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Be able to add, subtract, multiply and divide multidigit decimals
sans and with decimal points on paper, without a calculator.
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know how to read multidigit decimals aloud - there is site invention
here that will help place value comprehension before and after
decimal point.
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know how to measure and figure with fractions and mixed numbers.
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know how to add, subtract, divide and multiply fractions first in
special cases and then in general - if you do not know how raising
terms leads from special to general cases, here are ideas for
yourself and your student to see, and if not overwhelming, to master.
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Learn how to recognize primes and how to factor composite numbers
efficiently. The old square method (site name for it) emphasized here
fill a gap in course design by providing an easy and quick prime
factorization. How to use calculators in this is explained.
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Learn how to use prime factorization to find GCDs, LCDs and LCMs, and
with those skill how to add, compare, subtract, multiply and divide
fractions efficiently.
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Learn how to add, multiply, subract and divide signed numbers -
integers and rationals included. The how-TOs here are clearly put in
common or vulgar terms to make the hard easier.
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Learn how to evaluate arithmetic and algebraic expressions with a
format that does and record steps in manner that skill can be seen
and confirmed, or corrected.
Lessons to develop the above skills and allied may be based on site
arithmetic and number theory folders. The ability to count, figure and
measure well with whole numbers and fractions, while avoiding mistakes,
has take-home value. That sets the stage for care in work and study, and
shows how following rules and methods, step by step may lead to
observable, repeatable and reproducible results. The further skills above
have less and less take home value, but they are required in algebra.
Schools which teach algebra before or without consolidating arithmetic
skills are wasting the time of their students and teachers.
Before writing began I saw and sensed gaps in course design and delivery.
Site material reflect and support inductive principles for skill
development online in site Volume 1B, Mathematics
Curriculum Notes.
Still More Notes for Instructors
The two of the popular site pages are provided by
The 15 Times Table and The 20 Times Table. A format or
how to do and record figuring steps in way that the doer and others may
see and verify is one key to observable and verifiable skill
development. The two lessons on (i) Arithmetic Expression Evaluation
how to do and show work and on(ii) Formula Evaluation - how to
do and show work emphasize good format. The page arithmetic
videos - Real Player Format links to 50 or so legacy videos on
some, if not of the following topics.
Arithmetic and Number Theory Skills may be developed and strengthened by
the lessons or lesson ideas in the following folders. Each folder
consists of webpages alone or with flash player videos. When this site
author determines what video format will serve web-browser on apple pcs
and tablets, that format will follow.
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1 Decimal Place Value: We may write decimals in standard and
expanded form. Besides that the tongue-twisting, very amusing for
students, exercise of reading multidigit decimals in groups of three,
first without and then with decimal points may entertain and inform
young mathematics and older science students. These lessons may
develop and strengthen place value, and helping learners grasp the
size or magnitude range which decimals can describe. Avogrado's
number for instant is a mere 602 Sextrillions. The topics covered
here will likely take more time to read than to present in class.
Teachers in the UK and other big BILLION places, where decimal place
in groups of 6 may be customary, may employ the SI standard
international units, so that their students may read decimals in
smaller groups of three - the latter presumably being easier
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Arithmetic with Decimals. This folder includes 50 or so
webpages and videos to provide a practical command and a
thought-based development of decimal methods for counting and adding,
comparing and subtracting, for multiplication without and with
decimal points, and for long division with remainder or to several
decimal places. Exercises are included. The extent of the
thought-based development is likely to be overwhelming for most
students in class. Here skill and confidence may follow and have
greater take-home value if students learn decimal methods to get
repeatable and reproducible results, and in the process learn about
the domino effects of mistakes, and the need to avoid them. The
development here emphasizes that subtraction can be checked by an
addition, that long division with quotient and remainder can be
checked via a multiplication followed by an addition of the
remainder. When a check fails, students need to be told that the
error in their figuring lies between the start of their work and the
end of their check.
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Prime Factorization Skills. This folder offers an efficient,
self-contained exposition of composite and prime numbers, along with
a square rule method for quick recogniztion of primes, and quick
prime factorization or decomposition of non-primes. The number theory
practices here do not have immediate take home value, but they set
the stage for the efficient simplification of fractiosn and roots
which mathematical preparation for the calculus dependent college
programs - commerce, accounting, engineering, science and technology
require. Factoring by inspection of quadratics and higher degree
polynomials with integral coefficients may appreciate the counting
and generation of all factors of a given composite number.
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Remainder Arithmetic and Divisibility. This folder represents
optional material. It provides a numerical development of its topic
which gifted students may appreciate, and which instructor may
supplement in upper year, secondary algebra courses, by providing the
missing theory.
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Integers. This folder provides a thought-based developed of
arithmetic and number theory practices with integers. Two ways to
recognize whole numbers multiples of 11 are included, one with and
one without the use of integers.
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Fractions and Ratios. Raising terms is well-established
approach in class to introduce and justify methods for the addition,
subtraction and comparison of fractions. Raising terms approach is
extended here to introduce methods for division and multiplication of
fractions. How to add, subtract and multiply efficiently is also
described in 22 lessons. Three appendices describe the similarity and
differences between fractions and two and multi-term ratios.
Fractions may be identified with a two term ratio of a part to a
whole, but ratios of parts to parts, and mutli-term ratios are not
fractions.
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Arithmetic and Fractions with Units. This topic has take-home
value in the discussion of cost per unit, hourly rates and wages, and
in distance-speed-time calculation. Students in chemistry, physics
and even biology have to calculate with units and change units alone
or in rates given as "fractions". This topic with it simplification
of monomials given by products of numebers and units to various
powers provides an contextful alternative to student manipulation of
monomials in products of numbers and variables - the variables being
formal symbols without meaning for most. Finally, the discussion of
proportionality constants and the application of mathematics in model
problems involving physical measurement are best done with a command
of algebraic-like, arithmetic with units.
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Squares and Square Roots. These can be given exactly or
approximately with a calculator. But as part of the mathematical
preparation for college programs which require calculus mentioned
above, the exact expression of square and higher roots in a product
of whole numbers and roots of irreducible whole numbers. Here is
another use of prime factorization.
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Combinatorics - Trees Tables and Products. Counting how place
or spots there are in rectangular arrays provides one base for
calculating number of squares in a region. Via numerical examples,
the five videos in this folder compare, contrast and develop or
illustrate tree, tabulation and product methods for counting.
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LCM GCD and Euclid GCD Algorithm. The 17 videos in this folder
introduce numerical practices for calculating least common multiples
and greatest common divisor with a list method - elsewhere called
inspection, with prime factorization, and with Euclid's Algorithm.
All is done forwards and backward. The material has no take-home
value, but they help in efficient simplification of fractiosn and
roots which mathematical preparation for the calculus dependent
college programs - commerce, accounting, engineering, science and
technology require.
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For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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